Question

A factory produces a radioactive substance $A$ at a constant rate $R$ which decay with a decay constant $\lambda$to a stable substance. Find
(i) The number of nuclei of $A$ and.
(ii) Number of nuclei of $B$, at any time $t$ assuming the production of $A$ starts at $t=0$.
(iii) Also, find out the maximum number of nuclei of $A$ present at any time during its formation

Answer 1

$r-\lambda N=\frac{dN}{dt}$

$dt=\frac{dN}{R-\lambda N}$

Integrating on both sides.

${\int }_{t=0}^{t}dt={\int }_{N=0}^N\frac{dN}{R-\lambda N}$

$t={\int }_{N=0}^N\frac{dN}{\lambda (\frac{R}{\lambda }-N)}$

$t=\frac{1}{\lambda }={\int }_{N=0}^N\frac{dN}{(\frac{R}{\lambda }-N)}$

$t=\frac{1}{\lambda }-[ln|\frac{R}{\lambda }-N|]$

$t=\frac{-1}{\lambda }[ln[\frac{R}{\lambda }-N]-ln\left[\frac{R}{\lambda }\right]]$

$t=\frac{-1}{\lambda }\left[ln\left[\frac{R/\lambda -N}{R/\lambda }\right]\right]$

$R/\lambda e^{-\lambda t}=R/\lambda -N$

$N=R/\lambda [1-e^{-dt}]$